Why won't it stay on?

What happens before it shutting off itself?
Are you sure that the timer off isn't on?
Normally, the self off may happen because the TV goes into Protection Mode due to a failed circuit board.

back the carb screw out 1/4 turn.

Do the self check.

If you unplugged the set, plug it back, turn it on and allow to fail.

Press both the Menu and Input (input may be labeled device) buttons on the front of the set and Hold both for at least 5 seconds and release both.

Power light will begin to blink a two digit error code like 2 blinks, 2 blinks, pause and repeat it for a total of 5 times.

Let me know what error you find and I tell you the meaning and a possible answer.

Could be any of several things. If the crank sensor is marginal and not sending pulses, the fuel pump is shut off. This is for safety. The pump is energized for a few seconds as key is turned in preparation for starting then engine must rotate normally to keep the pump on. If it stops, in a few seconds the fuel is cut off.

One thing to check into would be the antitheft system. Not knowing what year or type of vehicle makes it hard to help. But like i said, engine shut down in 5 seconds or less is a symptom of a bad antitheft security system. This link explains.
http://vatspasslockpasskeysecurityhelp.yolasite.com/

check fuel pressure, should be 40 psi.
try 2 things

spray starting fluid in air intake to see if it will stay running. It's fue then.

Try turning key to on nd wait 5 seconds for system to build pressure and pump to shut off. Turn key off, wait for relay clcik noise or 10 seconds. then turn key to ON again and wait 5 seconds to build pressure again. Then start engine. If it starts better, you have a bad check valve in the fuel pump or a bad pressure regulator on the fuel rail (unlikely).

report back

The first couple of seconds of operation, most TVs are doing some self-test and if something has failed making full power on dangerous (for fire) they will shut down. Either a major component in the horizontal section has failed or the main power supply itself is defective.

Do the self check when power light goes off.

On front of set press and hold both the menu and input buttons for 5 seconds or longer and release both buttons.

Power light will blink a error code like: 2 blinks, 2 blinks, a pause and repeat about 4 more times.

Let me know what error code you find and I will tell you what it means.

If this information told you something rate this solution as very helpful as I do this here for free.

SD TECH

if it starts and warms up does it shut it self off or do you shut it off an then won't restartif it shuts self off may be runnining it self out of gas wwhen it shuts off try spraying a little starting fluid in breather and see if it will recrank if it does you have a fuel problem if it does not start you have an electicly problem such as a bad coil poss